8G Slides Ch 7 Systems.ppt

8G Slides - Chapter 7

Systems

7.3.1 - Graph Systems

of Equations

• Key Skill: WWBAT solve systems of

Slope-Intercept Form

• Key Skill: WWBAT rewrite equations in Slope-Intercept form (y=mx+b) by

Key Vocabulary

equation is shown as y=mx+b where ‘m’ is the slope and ‘b’ is the

y-intercept.

• For example, in the equation y = 3x – 4 the slope is 3 and the

Slope-Intercept Form

Slope-Intercept form (y=mx+b), where ‘y’ is alone

What if the equation

looks different?

• y + 1 = 2x - 4

What if the equation

looks different?

• y + 1 = 2x - 4

• What can we do to put the equation in y=mx+b form?

y + 1 = 2x - 4

Examples

Examples

2x + y = 7 -2x -2x y = -2x + 7

-3y + 9x = 15 -9x -9x -3y = -9x +15 ÷(-3) ÷(-3) y = 3x - 5

Key Vocabulary

• A system of equations is simply a group of two or more equations.

Importance of Systems

to compare different situations and find “break-even” points.

• Example: One insurance plan charges $1,000/month plus $100/doctor visit. Another charges $1,300/month but

Solution v Solution Set

x+1=x

• Some equations have a single solution, like x+1=2

• Some equations have two solutions, like x2=9

Example

y = x + 2 y= -x + 4

• In systems, any solutions for ‘x’ and ‘y’ must work for BOTH equations.

• There are several ways to find the solution. • The first we will examine is graphing both

y=x+2

Solutions

• ALL of the solutions for one equation can be found on one line, and ALL of the solutions for the other equation can be found on the other line.

• The only solution(s) to BOTH equations can be found where the lines meet!

Some Systems Have No

Solution

• Look at these equations:

y = 3x + 7 y = 3x - 2

Parallel lines

Classwork

7.3.2 - Problems

Involving Systems

Example

• 2 bookstores charge the same price for books.

• Gaslight Books charges a shipping fee of $3 per order plus $1 per book.

• Crimescene Books charges $5 per order plus $0.50 per book.

Solution Method

y is total shipping cost

x is number of books ($1/book) $3 is shipping cost for the order

• Crimescene: y = .5x + 5

y is total shipping cost

Graph of Solution

1 2 3 4 5 6 7 8 9 10 Books Ordered To ta l Sh ip p in g C os t Gaslight Crimescene

Interpreting the Solution

both stores when we order 4 books.

• Which is cheaper when we order fewer than 4 books?

Checking Your Solutions

correct.

7.3.3 - Substitution

Graphing Solutions

disadvantages:

– When numbers get very large

Substitution

Basic Example

Basic Example

y = 2x+1

Simply “substitute” the number 5 for the variable ‘x’ in the second equation:

Another Example

Another Example

x = y - 7

y = 3(y-7) - 5 Substitute (y-7) for ‘x’

y = 3y - 21 - 5 Distributive Property y = 3y - 26 Combine Like Terms

0 = 2y - 26 Subtract ‘y’ from both sides 26 = 2y Add 28 to both sides

13 = y Divide both sides by 2

More Examples

a = 3 - b

4a + b = 15

x = 2 - y

More Examples

4a + b = 15

4(3-b) + b = 15 12 - 4b + b = 15 12 - 3b = 15

-3b = 3 b = -1 a = 4

x = 2 - y

8y + x = 16

8y + (2-y) = 16 8y + 2 - y = 16 7y + 2 = 16

Word Problem Example

the cost of plain socks

Example

• Designer socks cost $1 more than twice the cost of plain socks

y = 2x + 1

• The price for 3 pair of plain socks and 1 pair of designer socks is $11.

Solution Using Substitution

y = 2x + 1 3x + y = 11

• If we know that y = 2x + 1, we can rewrite the second equation,

substituting 2x + 1 for y

3x + (2x + 1) = 11

Solution (continued)

3x + 2x + 1 = 11 No need for parens 5x + 1 = 11 Combine like terms -1 -1

5x = 10 Isolate variable ÷5 ÷5

Interpreting the Solution

$2 is the cost of the plain socks

• We must go back and solve for ‘y’ to get the cost of the designer socks

y = 2x + 1 y = 2(2) + 1 y = 5

Checking Your Solutions

solutions back into BOTH of your equations to prove you are correct.

Classwork

Substitution

(continued)

Example

equations in which neither ‘x’ or ‘y’ is alone.

Solution

6x - 2y = 29

First, change the first equation to get ‘y’ alone: -y = -2x + 9, then -y = 2x - 9

Now we can substitute 2x - 9 into the second equation for ‘y’

6x - 2(2x-9) = 29 Distribute and solve

6x - 4x + 18 =29 Make sure -2 times -9 = +18 2x + 18 = 29 Combine like terms

2x = 11

Key Skills

• In order to solve systems of equations algebraically, you must be able to:

– Manipulate equations to get ‘x’ or ‘y’ alone

– Use the distributive property with positive and negative numbers

Practice Day for Systems

• Starbucks plans a new blend and must find the right mix for a 6 lb batch that will sell for

$6.67/lb.

Starbucks

• Starbucks plans a new blend and must find the right mix for a 6 lb batch that will sell for $6.67/lb.

• The batch includes Kenya coffee, priced at $6/lb, and Sumatra coffee, priced at $8/lb.

• Step 1: Find two equations to make a system. x + y = 6

Starbucks

• Step 1: Find two equations to make a system.

x + y = 6

6x + 8y = 6(6.67)

Starbucks

• Step 1: Find two equations to make a system.

x + y = 6

6x + 8y = 6(6.67)

• Step 2: Change to y=mx+b format y = -x + 6

Starbucks

• Step 1: Find two equations to make a system.

x + y = 6

6x + 8y = 6(6.67)

• Step 2: Change to y=mx+b format

y = -x + 6

y = -(3/4)x + 5

Starbucks

Starbucks

– If x=4, then Starbucks needs 4 lbs of the Kenyan blend.

Starbucks

– If x=4, then Starbucks needs 4 lbs of the Kenyan blend.

– Then y=2 because x+y=6, so Starbucks needs 2lbs of the Sumatra blend.

• Check answer:

6(4) + 8(2) = 6(6.67) 24 + 16 = 40

7.3.4 - Elimination

What is the Goal?

What is the Goal?

• When we have a system of equations with two variables, what must we do to be able to solve the problem?

Example

• Look at the following system:

5x + 4 = 13 + 3y 2x = 3y

• What could we do to eliminate one

Solution

5x + 4 = 13 + 3y 2x = 3y

3x + 4 = 13 Subtract 2nd equation from 1st -4 -4 Now solve for ‘x’

3x = 9 ÷3 ÷3

Solution (continued)

We now must use the ‘x’ value to solve for y.

2x = 3y Use simpler of two equations 2(3) = 3y Insert known value

6 = 3y

÷3 ÷3 Solve for ‘y’

2 = y Solution for ‘y’

Checking Your Solutions

solutions back into BOTH of your equations to prove you are correct.

This is Big!!!!!

• You can actually add and subtract entire equations from one another!

Substitution: Not as easy

5x + 4 = 13 + 3y 2x = 3y

x = 1.5y Solve for ‘x’

5(1.5y) + 4 = 13 + 3y Substitute 7.5y + 4 = 13 + 3y Simplify

4.5y + 4 = 13 Solve for ‘y’ 4.5y = 9

y = 2 Solution for ‘y’

Key Vocabulary

• Elimination is an algebraic method to solve systems by adding or subtracting entire equations from one another in order to create a new equation with

Adding Equations

Adding Equations

3x + 5y = 10

5x - 5y = 14 Subtracting won’t help

8x = 24 But adding makes it easy

x = 3

3 (3) + 5y = 10 Then use ‘x’ to solve for ‘y’

Add or Subtract?

• If the variable we are eliminating has the SAME SIGN in both equations, we SUBTRACT.

Example: 5x + 4y = 25 3x + 4y = 11

• If the variable we are eliminating has DIFFERENT signs, we ADD the two equations

Word Problem

• John and Jen planted tomatoes and cucumber plants in their gardens.

Solution

• Let y=cost of cucumber plant 7x + 3y = 54

7x + y = 46 2y = 8

Elimination (cont.)

solve systems of equations when at least one equation must be first

More Complex Example

3x - 2y = 40

Example

7x - 4y = 100 Multiply the 2nd equation 3x - 2y = 40 by 2 and we get:

7x - 4y = 100 Subtract the equations 6x - 4y = 80

x = 20

Now You Try

10x + 9y = 19 8x + 6y = 24

Solutions

10x + 9y = 19 8x + 6y = 24

20x – 3y = 17 9x + 10y = 27

20x + 18y = 38 40x + 30y = 120

20x – 3y = 17 27x + 30y = 81

21y = 21 13x = 39

y = 1 and x = 1 x = 3 and y = 0 BOTH equations

must be

Word Problem

Word Problem

• A theatre sells 425 tickets for a total of $3,820. Adult tickets are $12 and kids tickets are $8. How many of each were sold?

a + c = 425 8a + 8c = 3400

12a + 8c = 3,820 12a + 8c = 3820 -4a = -420

What if the equation

looks different?

• y + 1 = 2x - 4

What if the equation

looks different?

y + 1 = 2x - 4

Examples

Examples

2x + y = 7 -2x -2x y = -2x + 7

3x - y = 3 -3x -3x

Examples

Examples

2x + y > 7 -2x -2x y > -2x + 7

3x - y > 3 -3x -3x

-y > -3x + 3 ÷(-1) ÷(-1)